Question: Suppose that $u$ is a vector in $\mathbb{R}^{3}$ of length 1, and let $w = u^{t}$ be the transpose of $u$. Let $R$ be the matrix $R = I - 2uw$. Explain why $R$ is called a reflection matrix, by explaining why, for any vector $v, wv = \langle u,v \rangle$, and explaining what $R$ does to the vectors on the line spanned by $u$, and what $R$ does to the vector perpendicular to $u$.

I've been spending a lot of time going over linear transformation and watching tutorials on transformation matrices. I found out that this matrix is actually called a 'householder reflection' but I couldn't find too much information about it in terms of the question asked.

I can prove that $R$ is orthogonal using $RR^{T}= I$. I'm not sure if that helps much.

I know if $\|\vec{u}\| = 1 $ then $uu^{T} = 1$

Because an orthogonal matrix preserves length and angles that means it limits the transformations to reflections and rotations.

Any vector that is perpendicular is left unchanged. Let's say $z$ is perpendicular to $u$ $$(I-2uu^{T})z = z-2u(u^{T}z)=z$$